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DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS
COLLEGE OF ENGINEERING AND TECHNOLOGY
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529-0247
PREDICTION OF RESPONSE OF AIRCRAFT PANELS SUBJECTED
TO ACOUSTIC AND THERMAL LOADS
By
Chuh Mei, Principal Investigator
Final Report
For the period December 1987 to March 1992
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665-5225
Under
NASA Research Grant NAG-I-S38
Dr. Steve A. Rizzi, Technical Monitor
ACOD-Structural Acoustics Branch
May 1992
https://ntrs.nasa.gov/search.jsp?R=19920015102 2020-07-30T08:04:47+00:00Z
SUMMARY
A unique cooperative graduate studies in structural acoustics and fatigue has been supported
for four years under NASA Grant NAG-I-838. The graduate educational program was jointly
established by the Old Dominion University, NASA Langley Research Center and the Air Force
Wright Laboratory in August 1989. Full and part-time graduate students earned masters and
doctoral degrees while working in the laboratories at NASA Langley Research Center, and a
spectrum of acoustic fatigue research was conducted. Research results documented in reports,
papers and theses were produced in the following sonic fatigue activities:
1. Influence of Transverse Shear Deformation on Large Amplitude Random Response of
Symmetric Composite Laminates,
2. Finite Element Free and Forced Vibrations of Rectangular Thin Composite Plates,
3. Finite Element Large Deflection Multi-Mode Random Response of Beam and Plate Structures
at Elevated Temperature,
4. Numerical Simulation of Large Deflection Random Vibration of Laminated Composite Plates
Using Finite Elements,
5. Finite Element Analysis of Thermal Buckling, Thermal Post-Buckling and Vibrations of
Thermally Buckled Composite Plates,
6. Thermal Environment and Response of Quartz-Lamp Heating System, and
7. Prediction of the Three-Dimensional Rectangular Duct using the Boundary Element Method.
This report presents a brief summary of the accomplishments achieved during the four-year
period. A list of publications, presentations and theses from this research grant is included.
TABLE OF CONTENTS
SUMMARY
1.
2.
.
.
.
.
.
8.
............................................... i
INTRODUCTION .............. • • • • • • • • . ................ 1
INFLUENCE OF TRANSVERSE SHEAR ON NONLINEAR RANDOM
VIBRATION .......................................... 2
NONLINEAR FREE, FORCED AND RANDOM VIBRATIONS USING FINITE
ELEMENTS .......................................... 7
3.1. Free and Forced Vibrations .............................. 7
3.2. Random Vibrations ................................... 8
3.3. Random Vibrations of Thermally Buckled Structures ............... 9
FINITE ELEMENT/NUMERICAL SIMULATION OF NONLINEAR RANDOM
RESPONSE OF COMPOSITE PLATES .......................... 18
THERMAL BUCKLING AND POST-BUCKLING, AND VIBRATIONS OF
THERMALLY BUCKLED COMPOSITE PLATES ................... 22
PREDICTION OF THE RADIANT FIELD PRODUCED BY QUARTZ HEATING
SYSTEMS ........................................... 29
BOUNDARY ELEMENT RESEARCH .......................... 35
REFERENCES ........................................ 39
PUBLICATIONS, PRESENTATIONS AND THESES .................. 42
9.1.
9.2.
9.3.
Publications ...................................... 42
Presentations ..................................... 44
Theses and Dissertations .............................. 45
ii
1. INTRODUCTION
This report summarizes the research accomplishments performed under the NASA Langley
Research Center Grant No. NAG-I-838, entitled "Prediction of Response of Aircraft Panels
Subjected to Acoustic and Thermal Loads," for the period December 16, 1987 to March 15, 1992.
The primary effort of this research project has been focused on the development of analytical
methods for the prediction of random response of structural panels subjected to combined and
intense acoustic and thermal loads. The accomplishments on various acoustic fatigue research
activities are described first, then followed by publications and theses.
2
2. INFLUENCE OF TRANSVERSE SHEAR
ON NONLINEAR RANDOM VIBRATION
The effects of transverse shear deformation and large deflection on random response of
symmetrically laminated composite rectangular plates were investigated the first time [i-3] in
the literature. The first-order shear deformation plate theory for laminated composite materials by
Yang, Norris and Stavsky [4] and the von Karman large deflection nonlinear strain-displacement
relations are used in the development of governing differential equations of motion. The
governing equations can be expressed in terms of the transverse deflection w and the Airy
stress function F [1] as
37(11 + I2) + UI(W) -- 0 (2.1)
- 2A_6F,_y +(2A_._ + A_6)F,z_uu -2A*16F,_yy u +A_lF, uyyy
.)
= W_xy -- W,xztO,yy
where
02W
I1 = p(t) - ph Ot 2
12 = F, yy w,z x +F,_z w,uy -2F, xy w,xy
and N and U 1 are the mathematical operators defined as
0 4 0 4 0 4 0 4
N = kl_y4__ q- k20x30------ _ + k30x20y 2 .-}- k4 OzOy-----304 0 2 02 02
+ ks-b-fi,+ k6b- x + + -
(2.2)
(2.3)
(2.4)
(2.5)
3
_4 (94
['1 = Dl1_ + 4D16 0x30y
04 06 06
+ D__,:_, + PI____ + P2 ax 5a-----_+ P3
06 36 06
+ t9o Ox2Ou_ + t96 OxOy-------_+ Pr Oy----g
04
-- + 2(D12 + 2D66)Ox.20y 2 + 4Do.6_
06 06+P4--
0,/:4 0y 2 Ox30113
04
OxOy 3
(2.6)
The excitation p(t) is assumed to be stationary, ergodic and Gaussian with zero-mean, the
coefficients k, and Pi in Eqs. (2.5 and 2.6) are functions of the laminate stiffness Ai*j and D,j
and are given in references [1-3]. The coefficients ki and Pi are due to the transverse shear and
they all go to zero if the shear deformation is neglected. The equations of motion are coupled,
nonlinear and of order sixth.
The Galerkin's method with a single-mode approximation is applied to yield a Duffing-
type forced modal equation. The equivalent linearization technique is then employed to obtain
the root-mean-square (RMS) maximum deflection and RMS maximum strains at various sound
spectrum levels (SSL). The transverse shearing strains or stresses, r_ and 7-y_, are obtained by
integration of the equilibrium equations of three-dimensional elasticity. Four cases of random
responses can be obtained from the analysis and they are: 1) linear, small deflection plate theory
without transverse shear deformation, 2) linear theory with shear, 3) nonlinear, large deflection
plate theory without shear and 4) nonlinear with shear. Boundary support conditions considered
are all four edges simply supported, all clamped, and simply supported on two opposing edges
and clamped on remaining two edges.
The nondimensional RMS maximum deflection and nondimensional RMS maximum strain
are shown in Figs. 1 and 2, respectively, versus plate length-to-thickness ratio (a/h) for three-
layer, cross-ply, square, graphite-epoxy laminate of 12x 12xh in. at SSL of 130 dB (Re 2x 10-5
N/m2). It is clear from the figures that the effects of transverse shear are considerable for plate
lengths are less than 30 times the thickness (a/h < 30). Linear small deflection theory with
4
sheardeformationwould give accurateresponsepredictionsfor moderatelythick plates. For
thin plates(a/h> 50), the linear and nonlinear solutions agree at low values of SSL, but disagree
at high values. The nonlinear large deflection theory with shear deformation neglected will give
accurate prediction for thin plates at high SSL values.
ILOd
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7
3. NONLINEAR FREE, FORCED AND RANDOM
VIBRATIONS USING FINITE ELEMENTS
It is logical that nonlinear free and forced vibrations of complex structures using finite
element will have to be first developed and well understood before any attempting on nonlinear
random vibrations.
3.1 Free and Forced Vibrations
Extension of the finite element methods to nonlinear free [5] and forced [6] vibrations of
beams was first presented by the principal investigator. Most recently, the finite element method
has been extended to the analysis of free and forced vibrations of arbitrarily laminated composite
plates [7, 8]. The system equation of motion for nonlinear forced vibrations of beam and plate
structures can be written as [7, 8]
"} + [Kll+ CK2]-[Pl/(w = 0 (3.1)
where [M], [N] and { W} are the mass, linear stiffness matrices and nodal displacement vector,
respectively. The matrices [K1] and [K2] are the first and second order nonlinear stiffness
matrices and depend linearly and quadratically upon nodal displacements, respectively. The
nonlinear stiffness matrices are due to the large deflections. The matrix [P] is the harmonic force
matrix which depends on the amplitude of vibration Wma x and the intensity of force excitation
Po. The nonlinear free vibration problem can be treated as a limiting case of the more general
forced vibration with Po or [P] = 0 in Eq. (3.1). The linearized updated-mode method with
nonlinear time function (LUM/NTF) approximation, which was introduced in solving nonlinear
structural-nonlinear aerodynamic panel flutter limit-cycle responses [9], is used for the free
vibration (backbone curve) and forced vibration response from Eq. (3.1).
The frequency ratios _-'/_tine_,r for S1 simply supported and C1 clamped [10] two-layered
angle-ply (30/-30), graphite-epoxy laminates of 12x 12×0.12 in. subjected to a uniform harmonic
8
forceof Po = 0 (free vibration), 0.1 and 0.2 are shown in Fig. 3. These results indicate that the
simply supported plate yields a larger nonlinear response than the clamped case.
The harmonic force matrix is developed for a uniformly distributed harmonic load over
the plate element. Application of the uniformly loaded finite element to the loading case for a
concentrated force is to allow the area of the loaded element to become smaller and smaller. This
is demonstrated for a concentrated force applied at the center of a simply supported square, three-
layer, cross-ply plate with S 1 [10] immovable inplane edges. The magnitude of the concentrated
force is equal to the same force for a plate under a uniformly distributed harmonic loading
of Po = 0.1 acting over the total plate area. The nonlinear response for a concentrated force
obtained with (//a) 2 = 1.0 percent is shown in Fig. 4. The difference in the frequency ratios is
less than 0.1% if a smaller loaded area of (l/a) 2 = 0.25 percent is used. The results show that
concentrated force is approximately two to three times as severe as the uniformly distributed
force for the case studied.
3.2 Random Vibrations
The finite element method is further extended to large deflection random response of beams
and plates using single or multimode by Chiang [11-13]. Geometrical stiffness matrices have
been developed for a beam and a rectangular plate element to account for the induced inplane
forces due to large deflections. RMS maximum deflection for a rectangular (15× 12×0.040 in.)
aluminum plate using single mode at SSL from 90 to 130 dB is shown in Fig. 5. Results using
classic equivalent linearization (EL) technique are also obtained to demonstrate the accuracy of
the finite element (FL) results.
Multiple-mode RMS maximum deflections (Wm_,x/V/_) at various SSL between 90 and
130 dB (Re. 2× 10-5 N/m 2) for a simply supported and a clamped aluminum beam of damping
ratio _ = 0.01 are shown in Table 1. To demonstrate the accuracy of the finite element (FE)
9
results,classicEL methodand theFokker-Planck-Kolmogorov(FPK) equationapproach(exact
solution to the forcedDuffing equation)arealso given.
3.3 Random Vibrations of Thermally Buckled Structures
A limited amount of investigations on structural response subjected to intense thermal and
acoustic loads exists in the literature [14]. Seide [15] was the first who studied large deflection
random response of thermally buckled simply supported beam. The well known classical
Woinowsky-Krieger large amplitude beam vibration equation was used. The Galerkin's method
and time domain numerical simulation were then applied for predicting beam random responses.
The Galerkin/numerical simulation approach was also applied recently by Vaicaitis [16, 17] to
simply supported metal and orthotropic composite rectangular plates. The classical von Karman
large deflection plate equations including the temperature and orthotropic property effects were
employed. The classic analytical investigations have been limited to simple beam [15] and
rectangular orthotropic plates [16, 17] with simply supported boundary conditions and uniform
temperature distributions.
Locke and Mei [18, 19] extended the finite element method for the first time to structures
subjected to thermal and acoustic loads applied in sequence. The thermal load considered is a
steady-state temperature distribution AT(x, y). The thermal post-buckling structural problem is
solved first to obtain the deflection and thermal stresses. The thermal deflection and stresses are
then treated as initial deflection and initial stresses in the subsequent random vibration analysis.
The Newton-Raphson iterative method is used in the thermal post-buckling analysis. For the
problem of nonlinear random vibration, the linear vibration mode shapes of the thermally buckled
structure are used to reduce the order of the system equations of motion to a set of nonlinear
modal equations of a much smaller order. The equivalent linearization technique is then used
to obtain iteratively the RMS response. Figures 6 and 7 show the excellent agreement between
10
the finite element and the Galerkin/numerical simulation results by Seide [15] for a simply
supported beam at a wide range of thermal post-buckling loads AT/ATc,. = 1.25, 2.00 and 7.25
and intense SSL between 90 and 130 riB.
To demonstrate the versatility of the ,_nite element method, a rectangular (12× 15x0.040 in.)
plate with clamped boundary conditions and subjected to a nonuniform temperature distribution
and at SSL from 90 to 120 dB (Re. 2× 10 -5 N/m 2) is analyzed. The RMS #-strain at the
midpoint of the long-edge and in the direction perpendicular to the edge is shown in Fig. 8. At
low SSL the maximum strain occurs at or near T/Tc, equal to one. However, at higher SSL,
the maximum strain is seen to occur at higher values of T/Tc,.
]1
§
n
In o
I"IOI
II
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I I j0 O0 tO
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q/xetuM g(]n.Llqdl_lV
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/ // /
// /
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s"/
.,_'
Ci'- ..... 7/ ci !c
IiC
0.1 ! , I ! !90 100 110 120 130
SOUND SPECTRUM LEVEL, dB
RE 2 x 10 .5 Pa
FEFE
Figure 5. RMS maximum plate deflection vs sound spectrum level using one mode
Table 1 Comparisonof RMS maximumdeflectionsfor simply supportedbeamand clampedbeamusingdifferent methods
14
FPK method EL method FE method
SSL single single three single three fivemode mode modes mode modes modes
Simply Supported90 0.3198 0.3194 0.3194 0.3212 0.3175 0.3174
100 0.8615 0.8456 0.8456 0.8648 0.8522 0.8520
110 1.8867 1.7938 1.7937 1.8645 1.8231 1.8222
120 3.6248 3.3843 3.3834 3.5569 3.4663 3.4637
130 6.6134 6.1325 6.1289 6.5217 6.4275 6.4174
Clamped
90 0.1005 0.1005 0.1006 0.1006 0.0980 0.0979
100 0.3155 0.3154 0.3156 0.3159 0.3078 0.3076
110 0.9392 0.9354 0.9357 0.9449 0.9231 0.9223
120 2.3549 2.2850 2.2845 2.3591 2.3113 2.3098
130 4.8720 4.5952 4.5883 4.8742 4.7361 4.7335
Threemodes-- mode 1, 3 and 5.
Five modes-- mode 1, 3, 5, 7 and 9.
t5
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18
4. FINITE ELEMENT/NUMERICAL SIMULATION OF NONLINEAR
RANDOM RESPONSE OF COMPOSITE PLATES
Large deflection random responses of composite plates using the finite element method
and time domain numerical simulation [20-22] have been investigated by Robinson. The finite
element nonlinear equations of motion for a composite plate subjected to random excitation are
derived by application of the principle of virtual work. A set of nonlinear algebraic equation:
is then obtained with a weighted-satisfaction of the governing equations of motion over a time
interval. The input Gaussian white excitation is generated by filtering a white noise process
through a linear filter with the desired cut-off frequency (6 kHz), drop-offs and amplifications.
The Wilson-0 parameter iteration algorithm is employed for longer time interval step. Five types
of response statistics are computed, they are the moments, spectra, auto-correlation, probability
distributions and peak probability distributions. Figures 9-11 show the maximum deflection and
strain spectra, and strain probability distributions of a clamped 12 × 15 x 0.048 in., graphite-epoxy
(El = 30 Mpsi, E2 = 1.2 Mpsi, G = 0.65 Mpsi, u12 = 0.25 and p = 1.449× 10-3 lb-s/in. 4) with
stacking sequence (90/45/-45/0)s at 123 and 153 dB. The spectra show the peak broadening
behavior of the nonlinear stiffness and the non-zero mean strain due to large deflection. The
strain spectrum at 123 dB also indicates a second peak due to membrane straining.
19
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=_
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°w.i
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! . t . 0 .
- . | .
I I I I
• 0 • •
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e-
lm
¢¢1
e_
22
. THERMAL BUCKLING AND POST-BUCKLING, AND
VIBRATIONS OF THERMALLY BUCKLED
COMPOSITE PLATES
The finite element system equations of motion for an arbitrarily laminated composite plate
undergoing small amplitude vibrations and subjected to temperature change AT(z, y) are derived
= {PAT}
[23, 24] and they are of the form as:
where [M] and [If] are the system mass and linear stiffness matrices, [KN,_XTJ is the geometric
stiffness matrix due to thermal membrane force vector {NAT}, and [N1] and [N2] are the
first-order and second-order nonlinear stiffness matrices due to large deflections. The matrices
I.V1] and IX2] are linearly and quadratically dependent on the system displacement vector {W},
respectively. The {PzT} is a steady-state thermal load vector due to temperature distribution
AT(x, y). A novel solution procedure has been developed in [23, 24] by considering the system
displacement {W} as the sum of a time-independent particular solution {W}_ and a time-
dependent homogeneous solution {W(t)} t with the assumption that the vibration amplitude
{W}t is small; that is
{w} = {w}, + {w(t)}, (5.2)
Substituting this into the system equations of motion, Eq. (5.1) can be resulted into two sets of
equations in the form as
([K]-[IfN/XT]+I[N1]s+3[N2]s){W}s
and
= {Par} (5.3)
[M]{I_'V} + ([If]- [KNaTI + [Nlls + [N2],){W}tt
= 0 (5.4)
23
where [N1]_ and IN2], denotethe first and second-orderstiffnessmatrices that are linearly
and quadraticallydependenton {W}_. Equation (5.3) is set of nonlinearalgebraicequations
which yieldsthe thermalpost-bucklingdeflectionunder thermalload {PAT}. It is solvedusing
incrementaltemperaturestepsanda Newton-Raphsoniterativeschemeto arriveat theconverged
post-bucklingdeflection{ W}s. Equation (5.4) is an basic eigenvalue problem, and it is evident
that the temperature ([KN,.XT]) and thermally buckled equilibrium position ([N1]s and [N2]s)
effects on the vibration characteristics have been both considered.
Figures 12 and 13 compare the thermal post-buckling deflections between the finite element
and classic analytical solutions. Figure 12 is for a clamped square aluminium plate and the
classical solution by Paul [25] is chosen because it uses 25 modal functions in the post-buckling
analysis and should be very accurate. Uniform and nonuniform temperature distributions
AT(x,y)=To 1-cos-- 1-cos (5.5)a
are considered. The finite element results compare very favorably with Paul's.
The post-buckling behavior of a simply supported, (+45/0/90)s, square (10 in.), graphite-
reinforced composite plate under uniform temperature is compared with Meyers and Hyer's
classic result [26]. As the plate lamination is symmetric, the critical temperature will be a
bifurcation point, at which the plate will buckle in either in the positive or negative direction,
and it will then follow one the deflection curves in the post-buckling range as the temperature
increases.
The influence of moderately large initial imperfections in deflection on thermal post-buckling
behavior can be easily considered in the finite element method. Figure 14 shows the maximum
thermal deflection of a simply supported, square (12 in.) graphite-epoxy (-30/30)2 laminate with
three different initial maximum deflections wt/h = 0.25, 0.50 and 0.75. The stiffness of the plate
is increased as the initial deflection is increased.
24
The changein thefundamentalfrequency_'/Woversustemperaturerise is shownin Fig. 15.
As expected,the fundamentalfrequencydecreasesto zero at the buckling temperature. The
effects that the initial imperfectionsin deflection hason the fundamentalfrequencyis also
shownin Fig. 15.Obviously,with any initial deflection,the platewill no longerbuckle andthe
frequencywill not decreaseto zerowith increasingtemperature.
2S
OXOX
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a Ns_
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26
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.... ¢..
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7_
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28
29
6. PREDICTION OF THE RADIANT FIELD PRODUCED
BY QUARTZ HEATING SYSTEMS
The ability to predict the radiant heat flux distribution produced by a reflected quartz heating
system is desirable for use in thermal-structural analyses. Therefore, a classical analysis technique
is developed for this purpose and proved very effective in the analysis of a single unreflected
quartz heater [27]. This classical analysis was extended to incorporate the effects of reflector.
Through this expansion process, serious limitations of the classical analysis were discovered
[28]. Therefore, it was decided that a more versatile and robust technique must be applied to
these more complicated heating systems.
A Monte Carlo simulation method [29] is thus developed for predicting the radiant heat
flux from a heating system. The heating system consists of tungsten-filament tubular quartz
lamps and individual or common (multi-lamp) reflectors. The simulation method allows for a
multitude of model refinements not amenable to the classical analysis developed earlier. Attempts
at correlating with experimental results revealed that a lamp must be treated as combination of
two sources, where the quartz envelope gains a majority of its energy from the emissions of the
corresponding filament. The source augmentation due to reflection and contributions from other
sources is modeled in the simulation.
The simulated results compare well with experimental measurements from the three systems
tests. The first system consists of single lamp of 25.4 cm lighted length. The lamp is placed
3.33 cm beneath a highly polished titanium reflector and 15.24 cm above the Gardon-type heat
flux sensor plane. The success of the Monte Carlo simulation shown in Fig. 16 is primarily
clue to the treatment of the lamp as two interacting sources. It also indicates that the inclusion
of those higher order effects is very important even for the simple cases where reflector and
source interaction is minimal.
30
Experimentalradiant heatflux were measuredfor a secondsystemconsistingof a single
(63.5cm) lamp anda parabolicreflectorand30.48cm abovethe sensorplane. The testdataare
shownin Fig. 17 in comparisonto the simulation predictions.
Figures 18and 19 show the radiantheatflux resultsof the third system,a single lamp or
six-lampwith a multiple six-lampreflector, respectively.
31
,.dg.r.,
_D
0.5
0.4
0.3
0.2
0.1
0 I I I I
-20 -15 -10 -5 0
Monte Carlo---Classical
+ + Experiment
I I 1 I
5 10 15 20
Y Posil.ion, cm
Figure 16. Transverse heat flux distribution for the single lamp with flat reflector
32
_
co
,--a
¢0
1.5 _
1 _
0.5 _
0
-20
Monte Carlo+ + ExperimenL
I I I f I l I f
-15 -10 -5 0 5 10 15 20
Y PosiLion, em
Figure 17. Transverse heat flux distribudon for the single lamp with parabolic reflector
33
,.2
0.6
0.5
0.4
0.3
0.2
0.1
I
m
E
m
0
-30
÷
+ +÷
÷
+
ExperimentMonte Carlo
÷1 I i I I I
-20 -10 0 10 20 30
Y Position, cm
Figure 18. Transverse heat flux distribution for the single lamp with multi-lamp reflector
34
¢9
,-d
5
4 _
3 _
2
1 _
+
0 I-30 -20
+ 4-
+
ExperirnentMonte Carlo
4-
I I I I I-10 0 10 20 30
Y Position, cm
Figure 19. Transverse heat flux distribution for the six-lamp heating system
35
7. BOUNDARY ELEMENT RESEARCH
The Boundary Element Method (BEM) is a good approximating technique that is widely used
in the solution of acoustic field problems. BEM needs less grid generation and computational
time, and is perfectly suited to study far-field acoustics. Over the past ten years, interest in
boundary element techniques has been very popular.
In the development of the newer supersonic and hypersonic flight vehicles, an increased need
for the analysis of acoustic fields and the response of structures to acoustic loads has become
necessary. Research in the analysis of these acoustic fields in ducts has dominated much of
the boundary element research. The first objective was to study the boundary element method
for acoustic problems in a two-dimensional field. Many types of problems with complicated
boundary conditions were investigated. All of the selected problems had exact solutions. These
exact solutions were used to validate the boundary element as an accurate approximating method.
Two types of elements were used: constant elements and linear elements. Both types of elements
gave very good approximations for all selected problems. As expected, the linear elements
produced less error than the constant elements for all of the problems For one problem, a
frequency graph was plotted for linear elements and compared with the exact solution shown
in Fig. 20 [30, 31]. The linear elements exactly predicted the pressure for wavenumbers less
than 2. For higher wavenumbers, the pressure values begin to deviate from the exact solution
slightly. From running different cases of varying wavenumbers, a very important idea was
learned about element discretization. There must be at least four elements per wavelength to
give accurate solutions.
The next step in research was to expand the domain problem to ducts with sudden area
changes [32]. Two basic muffler type problems were selected for comparison with the boundary
element method. In each case, constant and linear element results were compared with the exact
36
solutions.For bothproblems,theconstantandlinear resultsgavevery goodapproximationsfor
low wavenumbers.Sincethenumberof elementsdid notchange,it wasexpectedthattheresults
would deviatefrom the exact solutionasthe wavenumberincreased.This was evident in the
frequencyplots asthe wavenumberincreased.After validating the boundaryelementmethod
for suddenareachangesin ducts, anotherduct problemwas selectedwhereno exactsolutions
exist, Fig. 21. The transmittedpressurewasstudiedastheangleof themid-sectionwasrotated.
As the angle increased from 0 to 45 degree, it was expected that the transmitted pressure would
decrease exponentially. As seen in Fig. 21, the pressure did decrease very rapidly as the angle
increased. This type of duct problem was selected to show that the boundary element method
can handle a wide range of very complicated problems.
The next research step was to study three-dimensional duct problems with exact solutions.
This has been done for very simple problems. The computer code is now being updated to
handle more complicated three-dimensional problems. Research is also being done in the area
of combining finite and boundary element methods, finite element for modeling structures and
boundary element for modeling acoustics. This seems to be a very fast growing concept because
boundary elements and finite elements both offer special features that compliment each other.
By combining the two methods, more realistic and complicated problems can be handled.
37
i,i
t/)U3i,i
6
5
4
2
i
0
-I
-2
-4-
-5
-6m
0
EXACT SOLUTIONLINEAR ELEMENTS
I I I I
O.5 1 1.5 2
WAVE NUMBER (K-W/C)
Figure 20. Comparison of boundary element and exact solution
of a two-dimensional duct
38
THREE-SECTIONED RECTANGULAR DUCT
ACOUSllC
FIELD
p=l
dpRIGIO WALL _ =0
dn
/Ix
L_n."
f.m03ILl
EL
0.7
06
0.5
0.4
0.3
0.2
0.1
0 I0 b
<> oLINEAR ELEMENTSA A CONSTANT ELEMENTS
41
$$
$f i I I I I f
10 15 20 25 50 ,35 40 45
ANGLE (Degrees)
Figure 21. Transmitted pressure versus angle
39
8. REFERENCES
. Prasad, C. B. and Mei, C.: Effects of Transverse Shear on Large Deflection Random Re-
sponse of Symmetric Composite Laminates with Mixed Boundary Conditions. 30th Struc-
tures, Structural Dynamics and Materials Conference, Mobile, AL, April 1989, pp. 1716-1733.
-)Mei, C. and Prasad, C. B.: Effects of Large Deflection and Transverse Shear on Response
of Rectangular Symmetric Composite Laminates Subjected to Acoustic Excitation. J.
Composite Materials 23:606-639, 1989.
. Pl"asad, C. B. and Mei, C.: Effects of Large Deflection and Transverse Shear on Random
Response of Symmetric Composite Laminates with Various Boundary Conditions. Interna-
tional Conference on Advances in Structural Testing, Analysis and Design, Vedams Books
International, Bangalore, India, July 1990, pp. 138-144.
4. Yang, P. C., Norris, C. H. and Stavsky, Y.: Elastic Wave Propagation in Heterogeneous
Plates. Int. J. Solids Struct., 2:665-684, 1966.
5. Mei, C.: Nonlinear Vibration of Beams by Matrix Displacement Method. AIAA J., 10:355-357, 1972.
6. Mei, C. and Dechaumphai, K.: A Finite Element Method for Nonlinear Forced Vibrations
of Beams. J. Sound Vib., 102: 369-380, 1985.
. Chiang, C. K., Gray, C. E., Jr. and Mei, C.: Finite Element Large Amplitude Free and
Forced Vibrations of Rectangular Thin Composite Plates. In Vibration and Behavior of
Composite Structures, AD-Vol. 14, ASME Winter Annual Meeting, San Francisco, CA,
December 1989, pp. 53-63.
. Chiang, C. K., Gray, C. E., Jr. and Mei, C.: Finite Element Large Amplitude Free and
Forced Vibrations of Rectangular Thin Composite Plates. J. Vibration and Acoustics, ASME
Transactions, 113: 309-315, 1991.
. Gray, C. E., Jr., Mei, C. and Shore, C. P.: A Finite Element Method for Large-Amplitude
Two-Dimensional Panel Flutter at Hypersonic Speeds. 30th Structures, Structural Dynamics
and Materials Conference, Mobile, AL, April 1989, pp. 37-5 I. Also AIAA J., 29:290-298,1991.
10. Almroth, B. O.: Influence of Edge Conditions on the Stability of Axially CompressedCylindrical Shells. AIAA J., 4:134-140, 1966.
11. Mei, C. and Chiang, C. K.: A Finite Element Large Deflection Random Response Analysis
of Beams and Plates Subjected to Acoustic Loadings. AIAA 1 lth Aeroacoustic Conference,
Paper No. 87-2713, Sunnyvale, CA, October 1987.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
4O
Chiang, C. K. and Mei, C.: A Finite ElementLarge Deflection Multiple-Mode RandomResponseAnalysis of BeamsSubjectedto Acoustic Loading. Proceedingsof the 3rdInternationalConferenceon RecentAdvancesin StructuralDynamics, Institute of Soundand VibrationResearch,University of Southampton,July 1988,pp. 769-779.
Chiang, C. K.: A Finite Element Large Deflection Multiple-Mode Random ResponseAnalysis of Complex Panelswith Initial StressesSubjectedto Acoustic Loading. Ph.D.Dissertation,Old Dominion University, December1988.
Clarkson,B. L.: Reviewof Sonic FatigueTechnology.NASA Langley ResearchCenter,March 1990.
Seide,P. and Adami, C." Dynamic Stability of Beamsin a CombinedThermal-AcousticEnvironment.AFWAL-TR-83-3027,Wright-PattersonAFB, OH, October1983.
Vaicaitis,R. andArnold, R.: Time DomainMonteCarlo for NonlinearResponseandSonicFatigue. AIAA 13th AeroacousticsConference,Paper90-3918,Tallahassee,FL, October1990.
Vaicaitis,R.: NonlinearResponseandSonicFatigueof SurfacePanelsat ElevatedTempera-tures. Workshopon Dynamicsof CompositeAerospaceStructuresin SevereEnvironments,Sponsoredby ISVR and USAF, Southampton,UK, July 1991.
Locke, J. E. and Mei, C.: A Finite ElementFormulationfor the Large DeflectionRandomResponseof Thermally BuckledBeams.30 Structures,StructuralDynamicsand MaterialsConference,Mobile, AL, April 1989,pp. 1694-1706.Also AIAA J., 28:2125-2131, 1990.
Locke, J. E. and Mei, C.: A Finite Element Formulation for the Large Deflection Random
Response of Thermally Buckled Plates. AIAA 12th Aeroacoustics Conference, Paper 89-
1100, San Antonio, TX, April 1989.
Robinson, J. H. and Mei, C.: Influence of Nonlinear Damping on Large Deflection Random
Response of Panels by Time Domain Solution. AIAA 12th Aeroacoustics Conference. Paper
89-1104, San Antonio, TX, April 1989.
Robinson, J. H.: Finite Element Formulation and Numerical Simulation of the Large De-
flection Random Vibration of Composite Plates. Master's Thesis, Old Dominion University,December 1990.
Robinson, J. H: Variational Finite Element-Tensor Formulation for the Large Deflection
Random Vibration of Composite Plates. 32nd Structures, Structural Dynamics and Materials
Conference, Baltimore, MD, April 1991, pp. 3007-3015.
Gray, C. C.: Vibrations of Rectangular Plates with Moderately Large Initial Deflections at
Elevated Temperature using the Finite Element Method. 31st Structures, Structural Dynamics
and Materials Conference, Long Beach, CA, April 1990, pp. 2094-2100.
41
24. Gray, C. C. andMei, C.: Finite ElementAnalysisof ThermalPost-BucklingandVibrationsof ThermallyBuckledCompositePlates.32ndStructures,StructuralDynamicsandMaterialsConference,Baltimore, MD, April 1991,pp. 2995-3006.
25. Paul,D. B.: LargeDeflectionsof ClampedRectangularPlateswith Arbitrary TemperatureDistributions. AFFWAL-TR-81-3003,Vol. I, Wright-PattersonAFB, OH, February1982.
26. Meyers,C. A. andHyer, M. W.: Thermal Bucklingand Post-Bucklingof SymmetricCom-posite Plates,CompositeMaterialsConference,Michigan StateUniversity, East Lansing,MI, July 1990.
27. Turner, T. L. and Ash, R. L.: Prediction of the Thermal Environmentaland ThermalResponseof Simple Panelsto RadiantHeat. NASA TM-101660,October1989.
28. Turner,T. L. andAsh, R. L.: Analysisof theThermalEnvironmentandThermalResponseAssociatedwith Thermal Acoustic Testing. 31st Structures,Structural Dynamics andMaterialsConference,Long Beach,CA, April 1990,pp. 824-830.
29. Turner,T. L.: MonteCarlo Simulationof the RadiantField Producedby a Multiple LampQuartz HeatingSystem. 32nd Structures,StructuralDynamicsand Materials Conference,Baltimore, MD, April 1991,pp. 1393-1401.
30. Pates,C. S.: Predictionof the AcousticField in a Three-DimensionalRectangularDuctusingthe BoundaryElementMethod. 32ndStructures,StructuralDynamicsand MaterialsConference,Baltimore, MD, April 1991,pp. 2563-2568.
31. Pates,C. S. and Shirahatti,U. S.: Applicationof the BoundaryElementMethodto Predictthe Acoustic Field in a Two-DimensionalDuct. In Vibration Analysis w Analytical andComputational,DE -- Vol. 37, ASME 13th Conference on Mechanical Vibration and Noise,
Miami, FL, September 1991, pp. 253-257.
32. Pates, C. S.: Acoustic Analysis of Two-Dimensional Ducts with Sudden Area Changes
using the Boundary Element Method. 33rd Structures, Structural Dynamics and Materials
Conference, Dallas, TX, April 1992, pp. 360-368.
42
9. PUBLICATIONS, PRESENTATIONS AND THESES
9.1 Publications
1. Mei, C. and Prasad, C. B.: Effects of Non-Linear Damping on Random Response of Beams
of Acoustic Loading. J. Sound Vib. 117:173-186, 1987.
2. Prasad, C. B. and Mei, C.: Large Deflection Random Response of Initially Deflected
Cross-Ply Laminated Plates with Elastically Restrained Edges. 29th Structures, Structural
Dynamics and Materials Conference, Williamsburg, VA, April 1988, pp. 232-241.
3. Chiang, C. K. and Mei, C.: A Finite Element Large Deflection Multiple-Mode Random
Response Analysis of Beams Subjected to Acoustic Loading. Proceedings of the 3rd
International Conference on Recent Advances in Structural Dynamics, Institute of Sound
and Vibration Research, University of Southampton, July 1988, pp. 769-779.
4. Mei, C. and Prasad, C. B.: Effects of Large Deflection and Transverse Shear on Response
of Rectangular Symmetric Composite Laminates Subjected to Acoustic Excitation. J_
Composite Materials 23:606--639, 1989.
5. Locke, J. E. and Mei, C.: A Finite Element Formulation for the Large Deflection Random
Response of Thermally Buckled Beams. 30 Structures, Structural Dynamics and Materials
Conference, Mobile, AL, April 1989, pp. 1694--1706.
6. Prasad, C. B. and Mei, C.: Effects of Transverse Shear on Large Deflection Random
Response of Symmetric Composite Laminates with Mixed Boundary Conditions. 30th
Structures, Structural Dynamics and Materials Conference, Mobile, AL, April 1989, pp.
1716-1733.
7. Locke, J. E. and Mei, C.: A Finite Element Formulation for the Large Deflection Random
Response of Thermally Buckled Plates. AIAA 12th Aeroacoustics Conference, Paper
89-1100, San Antonio, TX, April 1989.
8. Robinson, J. H. and Mei, C.: Influence of Nonlinear Damping on Large Deflection Random
Response of Panels by Time Domain Solution. AIAA 12th Aeroacoustics Conference. Paper
89-1104, San Antonio, TX, April 1989.
9. Turner, T. L. and Ash, R. L.: Prediction of the Thermal Environmental and Thermal
Response of Simple Panels to Radiant Heat. NASA TM-101660, October 1989.
10. Mei, C., Wolfe, H. F. and Elishakoff, I.: Vibration and Behavior of Composite Structures.
Proceedings of the Symposium ASME Winter Annual Meeting, San Francisco, CA, Decem-
ber 1989.
11.
12.
3.
14.
15.
16.
17.
"18.
19.
20.
43
Shirahatti, U. S., Crocker, M. J. and Peppin, R. J.: Sound Power Determination from Sound
Intensity -- Results of the ANSI Round Robin Test. INTER-NOISE 89, Newport Beach,
CA, December 1989, pp. 1029-1034.
Shirahatti, U. S. and Crocker, M. J.: Sound Power Determination from Sound Intensity
ANSI Data Quality Indicators. INTER-NOISE 89, Newport Beach, CA, December 1989,
pp. 1035-1040.
Chiang, C. K., Gray, C. E., Jr. and Mei, C.: Finite Element Large Amplitude Free and
Forced Vibrations of Rectangular Thin Composite Plates. In Vibration and Behavior of
Composite Structures, AD-Vol. 14, ASME Winter Annual Meeting, San Francisco, CA,
December 1989, pp. 53-63.
Shirahatti, U. S. and Crocker, M. J.: Studies on the Second Power Estimation of a Noise
Source using the Two-Microphone Sound Intensity Technique. International Congress on
Recent Developments in Air and Structure-Bone Sound and Vibration, Auburn University,
AL, March 1990, pp. 49-56.
Shirahatti, U. S., Crocker, M. J. and Bokil, V. B.: Hartley Transform -- An Elegant Tool for
Sound and Vibration Analysis. International Congress on Recent Developments in Air and
Structure-Borne Sound and Vibration, Auburn University, AL, March 1990, pp. 193-200.
Zubair, M., Shirahatti, U. S., Jackson, T. L. and Grosch, C. E.: Solutions of Acoustic Field
Problems using Parallel Computers. International Congress on Recent Developments in Air
and Structure-Borne Sound and Vibration, Auburn University, AL, March 1990, pp. 903-908.
Turner, T. L. and Ash, R. L.: Analysis of the Thermal Environment and Thermal Response
Associated with Thermal Acoustic Testing. 31st Structures, Structural Dynamics and
Materials Conference, Long Beach, CA, April 1990, pp. 824-830.
Gray, C. C.: Vibrations of Rectangular Plates with Moderately Large Initial Deflections
at Elevated Temperatures using the Finite Element Method. 31st Structures, Structural
Dynamics and Materials Conference, Long Beach, CA, April 1990, pp. 2094-2100.
Prasad, C. B. and Mei, C.: Effects of Large Deflection and Transverse Shear on Random
Response of Symmetric Composite Laminates with Various Boundary Conditions. Interna-
tional Conference on Advances in Structural Testing, Analysis and Design, Vedams Books
International, Bangalore, India, July 1990, pp. 138-144.
Locke, J. E. and Mei, C.: A Finite Element Formulation for the Large Deflection Random
Response of Thermally Buckled Beams. AIAA J., 28:2125-2131, 1990.
*Received the Jefferson Goblet Student Paper Award.
44
21. Turner,T. L.: Monte CarloSimulationof the RadiantField Producedby a Multiple LampQuartz HeatingSystem. 32ridStructures,StructuralDynamicsand Materials Conference,Baltimore, MD, April 1991,pp. 1393-1401.
,.22.Pates,C. S.: Predictionof the Acoustic Field in a Three-DimensionalRectangularDuctusing the BoundaryElementMethod. 32ndStructures,StructuralDynamicsand MaterialsConference,Baltimore,MD, April 1991,pp. 2563-2568.
23. Robinson,J. H.: VariationalFinite Element-TensorFormulation for the Large DeflectionRandomVibrationof CompositePlates.32ndStructures,SmacturalDynamicsandMaterialsConference,Baltimore,MD, April 1991,pp. 3007-3015.
24. Gray,C. C. andMei, C.: FiniteElementAnalysisof Thermal Post-Bucklingand Vibrationsof ThermallyBuckledCompositePlates.32ndStructures,StructuralDynamicsandMaterialsConference,Baltimore, MD, April 1991,pp. 2995-3006
25. Petyt, M., Wolfe, H. W. andMei, C.: Proceedingsof the Fourth InternationalConferenceon RecentAdvancesin StructuralDynamics,Institute of Sound and Vibration Research,University of Southampton,ElsevierSciencePublisher,July 1991.
26. Chiang,C. K., Gray, C. E., Jr. and Mei, C.: Finite ElementLarge Amplitude FreeandForcedVibrationsof RectangularThin CompositePlates.J. Vibration and Acoustics, ASME
Transactions, 113: 309-315, 1991.
27. Pates, C. S. and Shirahatti, U. S.: Application of the Boundary Element Method to Predict
the Acoustic Field in a Two-Dimensional Duct. In Vibration Analysis -- Analytical and
Computational, DE - Vol. 37, ASME 13th Conference on Mechanical Vibration and Noise,
Miami, FL, September, 1991, pp. 253-257.
28. Turner, T. L. and Ash, R. L.: Prediction of Quartz-Lamp Heating System Performance using
Monte Carlo Techniques. Submitted to J. Heat Transfer.
9.2 Presentations
1. Mei, C.: On Analytical Methods of Structures Subjected to Thermal and Acoustic Loads.
Workshop on Analysis, Design and Testing of Structures Subjected to Thermal and Acoustic
Loads, Institute of Sound and Vibration Research, Southampton, July 1988.
. Mei, C.: Finite Element/Equivalent Linearization Nonlinear Random Vibration of Structures
at Elevated Temperatures. Workshop on Hypersonic Acoustic Loads and Response, NASA
Langley Research Center, Hampton, VA, October 1989.
**3.
45
Gray, C. C.: Vibrations of Rectangular Plates with Moderately Large Initial Deflections
at Elevated Temperatures using Finite Element Method. AIAA Regional Student Paper
Competition, Virginia Tech, Blacksburg, VA, April 1990.
4. Mei, C.: Analytical Methods and Development Activities in Sonic Fatigue and Panel
Flutter at ODU. Workshop on Dynamics of Composite Aerospace Structures in Severe
Environments. Institute of Sound and Vibration Research, Southampton, July 1991.
9.3 Theses and Dissertations
1. "The Effects of Nonlinear Damping on the Large Deflection Response of Smactures Subjected
to Random Excitation," by Prasad Chunchu Bhavani, Ph.D., May 1987.
2. "Effect of Boundary Conditions on Dynamic Response of Rectangular Plates," by Terry K.
Brewer, MS, May 1988.
3. "A Finite Element Formulation for the Large Deflection Random Response of Thermally
Buckled Structures," by James E. Locke, Ph.D., August 1988.
4. "A Finite Element Large Deflection Multiple-Mode Random Response Analysis of Complex
Panels with Initial Stresses Subjected to Acoustic Loading," by Chiyun-Kwei Chiang, Ph.D.,
December 1988.
5. "Finite Element Formulation and Numerical Simulation of the Large Deflection Random
Vibration of Laminated Composite Plates," by Jay H. Robinson, MS, December 1990.
6. "Finite Element Analysis of Thermal Post-Buckling and Vibrations of Thermally Buckled
Composite Plates," by Charles C. Gray, MS, May 1991.
7. "Monte Carlo Simulation of the Radiant Field Produced by Quartz Heating System," by
Travis L. Turner, MS, August 1991.
8. "Acoustic Analysis of Two-Dimensional Ducts with Sudden Area Changes using the Bound-
ary Element Method," by Carl S. Pates, III, ME, December 1991.
**Received the Second Place Award
